Unified a priori error estimate and a posteriori error estimate of CIP-FEM for elliptic equations

Jianye Wang; Rui Ma

doi:10.4208/aamm.2014.m834

Unified a priori error estimate and a posteriori error estimate of CIP-FEM for elliptic equations

Jianye Wang^*, Rui Ma

^*Corresponding author for this work

Peking University

Research output: Contribution to journal › Article › peer-review

Abstract

This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Comparedwith the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower H^1+s weak regularity under consideration, where 0 ≤ s ≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Ćea Lemma for conforming finite element methods can not be applied immediately when 0 ≤ s ≤ 1/2. To overcome this difficulty, our main idea is introducing an auxiliary C1 finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

Original language	English
Pages (from-to)	517-535
Number of pages	19
Journal	Advances in Applied Mathematics and Mechanics
Volume	8
Issue number	4
DOIs	https://doi.org/10.4208/aamm.2014.m834
Publication status	Published - 1 Aug 2016
Externally published	Yes

Keywords

Continuous interior penalty
Finite element methods
Posteriori error analysis
Priori error estimate

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10.4208/aamm.2014.m834

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abstract = "This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Comparedwith the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower H1+s weak regularity under consideration, where 0 ≤ s ≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and {\'C}ea Lemma for conforming finite element methods can not be applied immediately when 0 ≤ s ≤ 1/2. To overcome this difficulty, our main idea is introducing an auxiliary C1 finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.",

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AB - This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Comparedwith the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower H1+s weak regularity under consideration, where 0 ≤ s ≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Ćea Lemma for conforming finite element methods can not be applied immediately when 0 ≤ s ≤ 1/2. To overcome this difficulty, our main idea is introducing an auxiliary C1 finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

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Unified a priori error estimate and a posteriori error estimate of CIP-FEM for elliptic equations

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Keywords

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